Visualizing quaternions: An explorable video series (2018)

85•uncircle•6mo ago

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dang•5mo ago

Related. Others?

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gnabgib•5mo ago

Show HN: Visualizing the math that powers 3D character animation (238 points, 2022, 53 comments) https://news.ycombinator.com/item?id=31724613

calebm•5mo ago

Wow, this is very cool.

JKCalhoun•5mo ago

Very cool.

On MacOS: no audio on Safari, worked in Chrome though.

I was odd to me to see that Ben Eater was involved. His talents are broad.

srean•5mo ago

Questions for mathematicians out here.

Is there such a thing as quaternion analysis -- calculus of functions from quaternions to quaternions.

What would be their key theorems ? What would be the analogue of conformal mappings, if any ?

Any book recommendations would be gratefully appreciated.

Koshkin•5mo ago

I am not familiar with this area, but see, e.g. https://link.springer.com/book/10.1007/978-3-0348-0622-0

There are also these notes: https://www.geometrictools.com/Documentation/Quaternions.pdf

srean•5mo ago

Thanks a bunch

xeonmc•5mo ago

A quaternion encodes uniform scaling + rotation. The logarithm of a quaternion is its axis-angle-nepers form, and vice versa.

    quat = sqrt( exp( nepers + radians * <axis> ) )

So I think with this exponential map, the rest of its calculus can be extended from that.

srean•5mo ago

Heard the word 'nepers' after many decades. Are you by any chance an Electrical major ?

Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.

Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions

jacobolus•5mo ago

Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations. The 2 dimensional case is special.

See: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...

srean•5mo ago

Yes of course, but I am curious about any interesting structures that functions from quaternion to quaternion may possess. I used conformal mapping as an example of an interesting structure. I could have used Cauchy Riemann as another example.

jacobolus•5mo ago

You're probably looking for something like Sudbery 1977,

https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana...

(published 1979, doi: 10.1017/S0305004100055638)

srean•5mo ago

Fantastic. Thanks for the reference.

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Visualizing quaternions: An explorable video series (2018)

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